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2
votes
2answers
524 views

A problem about posets similar to Suslin's problem

Suslin's problem is: Given a complete dense linear order without endpoints, if it has the ccc must it be isomorphic to $\mathbb{R}$? The answer is that it's independent of ZFC. The related ...
3
votes
2answers
705 views

Does order isomorphism of linear extensions of two partially ordered sets imply order isomorphism of themselves?

Consider two partially ordered sets $A = \{a< b,a< c\}$, $B=\{x< z,y< z\}$. Their linear extensions (here we allow equality in linear extensions) for $A, B$ are $$A_L=\{A_1=\{a< b< ...
0
votes
1answer
205 views

Is every countable well-order embeddable in \mathbb{R}? [duplicate]

Possible Duplicate: Order types of positive reals The title is self-explanatory.
3
votes
2answers
238 views

What do you call a lattice whose meet operation preserves disjointness of subsets?

To make my question more precise and compact (and probably more intuitive), let me define the following: A subset $S$ of a lattice is mutually disjoint if for each $x \in S$, $\bigvee(S - \lbrace x ...
1
vote
0answers
323 views

Functoriality of a standard integral domain construction.

The evident forgetful functor from fields to integral domains has a left adjoint, namely the construction of the quotient field for a given integral domain. Another standard construction is taking the ...
10
votes
1answer
453 views

Partial word orders on groups

This is a followup question related to this question. Recall that a left-invariant partial order on a finitely generated group $G$ is called a partial word order if for every $a\le b\le c$ we have ...
6
votes
2answers
291 views

orders and length functions on finitely generated groups

Let $G$ be a finitely generated group with the natural word length function ($|x|$ is the length of the shortest word in generators of $G$ representing $x$). We call a partial left invariant order ...
8
votes
0answers
268 views

Higher-order dimension in posets: a reference request

Let $P = (X, \le)$ be a partially-ordered set. Then the dimension of $P$ is the minimum number of total orders over $X$ whose intersection yields $P$. Alternately, the dimension of $P$ is the minimum ...
0
votes
2answers
295 views

Is this a pre-ordered commutative semigroup?

Motivation I'm studying an approach to axiomatic thermodynamics based on the notion of commutative semigroup $(S,+)$ with a preorder relation $\to$ on $S$. In other words, $S$ is non-empty set, the ...
3
votes
0answers
375 views

a poset with small “cycles”

(a followup to this recent question) I noticed the following curious property of a poset (which I strongly believe to be a lattice, I'm still trying to prove that...): Suppose that $z$ is covered by ...
15
votes
3answers
3k views

Proving that a poset is a lattice

I discovered experimentally that a certain finite poset (sorry, I cannot give its definition here) seems to be in fact a (non-distributive, non-graded) lattice. The covering relations are reasonably ...
3
votes
4answers
1k views

What are examples of ordered fields that do not have the Archimedean Property?

Are the computable numbers one example?
3
votes
2answers
439 views

Arithmetic of ordered sets more general than ordinals

Motivation. Having read about infinite time Turing machines and ω-languages, I was thinking about more general notions of languages and “computation time”. Languages over strings of length ...
28
votes
4answers
2k views

A principle of mathematical induction for partially ordered sets with infima?

Recently I learned that there is a useful analogue of mathematical induction over $\mathbb{R}$ (more precisely, over intervals of the form $[a,\infty)$ or $[a,b]$). It turns out that this is an old ...
19
votes
4answers
999 views

When does a Galois connection induce a topology?

Let $(X,\leq)$ and $(Y,\leq)$ by partially ordered sets. Recall that a(n antitone) Galois connection between $X$ and $Y$ is a pair of order-reversing maps $\Phi: X \rightarrow Y, \ \Psi: Y ...
-3
votes
1answer
213 views

not sure what these basic FO symbols mean [closed]

I know some basic logic symbols, but i'm not sure what this formula means: Fragment: http://i35.tinypic.com/14jt1n9.png Full paper: http://www.newton.ac.uk/preprints/NI07003.pdf in particular, what ...
3
votes
0answers
194 views

An elegant formulation for typed sets

Fix a poset $T$, which we'll think of as a set of "types," interpreting $a \leq b$ as "$a$ is more general than $b$." Construct a category of TSet as follows. Objects: Pairs ($X$, $\tau : X ...
5
votes
2answers
431 views

Fraissé limit of the finite linear orderings

Hodges in his Shorter Model Theory promises to show "in what sense the finite linear orderings 'tend to' the rationals rather than, say, the ordering of the integers" (p. 160). After going through his ...
1
vote
1answer
563 views

The poset of k-small downward-closed subposets of a poset P is k-filtered when k is a regular cardinal?

Let $\kappa$ be a cardinal, and let $P$ be a poset. Let $\mathcal{P}_\kappa(P)$ denote the poset of $\kappa$-small subposets of $P$ and let ...
3
votes
2answers
197 views

Finite categories and partial orders

I'm studying category theory for the first time in a very succint book for computer scientists (I'm not actually a computer scientist, I'm a physicist, but my interest in cat theory is related to ...
3
votes
0answers
180 views

How linearly independent are the obvious combinatorial invariants of a Bruhat interval?

Let $[u, v]$ be a Bruhat interval in some Coxeter group. Let $I$ be the set of all Bruhat intervals. I am interested in functions $I \to \mathbb{Z}$ which are invariant under poset isomorphisms. ...
7
votes
1answer
283 views

Monotonic maximal chains in a Coxeter group

Let $(W, S)$ be a Coxeter system, and let $T = \bigcup_{w \in W, s \in S} wsw^{-1}$. Associated to every element $t \in T$ is a unique positive root $\alpha_t \in \Phi^{+}$ considered as a vector in ...
9
votes
1answer
439 views

Any further applications of Freudenthal's 1936 Spectral Theorem ?

Seemingly completely forgotten, back in 1936, the Dutch mathematician Freudenthal, quite well known at the time, proved his so called Spectral Theorem, see chapter 6 in Luxemburg & Zaanen : Riesz ...
15
votes
3answers
848 views

Subposets of small Dushnik-Miller dimension

The Dushnik–Miller dimension of a partial order $(P,{\leq})$ is the smallest possible size $d$ for a family ${\leq_1},\ldots,{\leq_d}$ of total orderings of $P$ whose intersection is ${\leq}$, ...
0
votes
0answers
171 views

Vector-valued valuations on lattices

There's a fair amount of work on valuations on (modular) lattices, by which I mean functions $v : \mathcal{L} \rightarrow R$ that satisfy the modular expression $$v(x) + v(y) = v(x \wedge y) + v(x ...
12
votes
5answers
1k views

Explicit ordering on set with larger cardinality than R

Is it possible to construct (without using Axoim of Choice) a totally ordered set S with cardinality larger than $\mathbb{R}$? Motivation: A total ordering is often called a “linear ordering”. I have ...
28
votes
2answers
679 views

What is the minimal size of a partial order that is universal for all partial orders of size n?

A partial order $\mathbb{B}$ is universal for a class $\cal{P}$ of partial orders if every order in $\cal{P}$ embeds order-preservingly into $\mathbb{B}$. For example, every partial order ...
3
votes
2answers
1k views

Cyclic order relation in Zn

The ring Zn:={0,1,..,n-1} under addition and multiplication modulo n. Suppose a,b,c,x $\in$ Zn are nonzero and the cyclic order R(a,b,c) holds, then under what conditions does R(ax,bx,cx) hold ?
8
votes
4answers
808 views

Order types of positive reals

Suppose one has a set $S$ of positive real numbers, such that the usual numerical ordering on $S$ is a well-ordering. Is it possible for $S$ to have any countable ordinal as its order type, or are the ...
6
votes
1answer
868 views

Correspondence between functions on a set and “states” on its power set

Let $L$ be the poset (ordered by set inclusion) that is the power set of some set $X$. A state is a function $s:L \rightarrow [0,1]$ satisfying i) for {$p_1,p_2,...$}, $p_i \in L$ a pairwise ...
-2
votes
1answer
640 views

Terminology: Lexicographical order [closed]

I would like to order a group of things by a set of rules of decreasing precedence. Please critique this sentence to help illustrate that: We define a lexicographical ordering for sheep by looking at ...
3
votes
4answers
479 views

(a,b) ≤ (a',b') iff b ≤ a' or (b' = b and a ≤ a') where a≤b and a'≤b'. Has any one seen this order?

I have been mucking round with orders, and this is the order that I found I needed. I would like to know if it is defined somewhere else, it is kind of difficult to search for such things. *EDIT* ...
23
votes
3answers
725 views

Does the exact pair phenomenon for partial orders occur in your area of mathematics?

Suppose that I have a partial order P and an increasing sequence $a_0< a_1<a_2<\cdots$ of elements of $P$. A pair of elements (b,c) from P is said to be an exact pair for this sequence, if ...
27
votes
9answers
2k views

How many groups of size at most n are there? What is the asymptotic growth rate? And what of rings, fields, graphs, partial orders, etc.?

Question. How many (isomorphism types of) finite groups of size at most n are there? What is the asymptotic growth rate? And the same question for rings, fields, graphs, partial orders, etc. ...
10
votes
5answers
3k views

infinite permutations

This question is related to this one: Continued fractions using all natural integers. Suppose we have the set of natural numbers $N$ with order and we perform permutation on it. So we obtain the same ...
12
votes
1answer
810 views

Kuratowski closure-complement problem for other mathematical objects?

The original Kuratowski closure-complement problem asks: How many distinct sets can be obtained by repeatedly applying the set operations of closure and complement to a given starting subset of a ...
5
votes
3answers
1k views

Constructing a metric over a lattice

Consider a lattice $({\cal L}, \wedge, \vee)$ with an antimonotonic function $f: {\cal L} \rightarrow {\mathbb R}$ defined on it (i.e $x \preceq y \implies f(x) \ge f(y)$). $f$ is said to be ...
-3
votes
2answers
646 views

Weak partitioning vs. strong partitioning

Let $U$ is a complete lattice with least element 0. Weak partitioning is a collection $S$ of nonempty subsets of $U$ such that $\forall x\in S: x\cap\bigcup(S\setminus\{x\})=0$. Strong partitioning ...
2
votes
4answers
652 views

Filter-closed vs. chain-closed

Let A is a complete lattice. I call a subset $S$ of A filter-closed when for every filter base $T$ in $S$ we have $\bigcap T\in S$. (A filter base is a nonempty, down directed set.) I call a subset ...
3
votes
3answers
224 views

Name for “lower/upper bounds” of arbitrary relations?

Given a partial order $R_{\leq}$ over a set $D$, the set of upper bounds under $R$ of a subset $S$ of $D$ is commonly defined as $\{ y \in D | \ \forall x\in S, x R y \}$. (The set of lower bounds of ...
5
votes
2answers
613 views

Reconstruction puzzles

[Added: This is a follow-up of an earlier post.] Consider the following "reconstruction puzzle", stated informally: Given a concrete poset, e.g. the poset of undirected unlabeled finite graphs ...
7
votes
1answer
523 views

Does ⋄ generate all De Morgan algebras?

(Asked by Nathaniel Hellerstein on the Q&A board at JMM) This question is about De Morgan algebras (see also Wikipedia), which are something like Boolean algebras, but with a different weaker ...
4
votes
2answers
437 views

Is every lattice the fixed-point set of an order endomorphism of ⋄^n?

(Asked by Nathaniel Hellerstein on the Q&A board at JMM) Let ⋄ be the 4 element lattice τ / \ i j \ / f Is every lattice isomorphic to the fixed point lattice of some ...
6
votes
2answers
389 views

closure of separative quotients

Does there exist a partial order, nontrivial for forcing, that is countably closed, but whose separative quotient is not countably closed? Supposing the answer is yes, then is there a partial order, ...
17
votes
2answers
703 views

Which are the rigid suborders of the real line?

Which are the rigid suborders of the real line? If A is any set of reals, then it can be viewed as an order structure itself under the induced order (A,<). The question is, when is this structure ...
1
vote
1answer
152 views

The proper name for a kind of ordered space [closed]

I'm trying to find the correct term for a specific kind of totally ordered space: Let $S$ be a totally ordered space with strict total order $<$. Property: For any two $s_{1}$ and $s_{2}$ in $S$ ...
2
votes
3answers
519 views

Countable atomless boolean algebra covered by a larger boolean algebra

Suppose Q is an atomless countable boolean algebra, and B is an arbitrary atomless boolean algebra. Q is unique modulo isomorphisms. There is a subalgebra in B that is isomorphic to Q. There is ...
6
votes
2answers
985 views

Ordinals that are not sets

The class of all ordinal numbers $\mathbf{Ord}$, aside being a proper class, can be thought of an ordinal number (of course it contains all ordinal numbers that are sets, not itself). Then one could ...
22
votes
2answers
721 views

How much choice is needed to show that formally real fields can be ordered?

Background: a field is formally real if -1 is not a sum of squares of elements in that field. An ordering on a field is a linear ordering which is (in exactly the sense that you would guess if you ...
3
votes
2answers
327 views

Semilattices in atomless boolean algebras

Let S be a bounded semilattice without maximal elements. Can we always construct an atomless boolean algebra B, containing S as a subsemilattice, such that S is cofinal in B-{1}? That is, for every ...